Consider a population evolving as a discrete-time supercritical multi-type Galton--Watson process. Suppose we run the process for $T$ generations, then sample $k$ individuals uniformly at generation $T$ and trace their genealogy backwards in time. In the limiting regime as $T \rightarrow \infty$, the expected behaviour of the sample's ancestry has been analysed extensively in the single-type case and, more recently, for multi-type processes in the critical case. In this paper, we present a formula for the distribution function of the generation $t$ of the most recent common ancestor in terms of the limiting distribution of the normalised population size. In addition, we provide effective bounds for the decay of this distribution function to 1 in terms of the harmonic moments of the population size at generation $t$. In order to better understand the behaviour of these harmonic moments, we use a multi-type generalisation of the Harris--Sevastyanov transformation to express harmonic moments at generation $t$ in terms of moments of the transformed process at the first generation. We present numerical results demonstrating that it is possible to approximate the coalescence time distribution effectively in practical settings.

Conversational interviews are commonly used to complement structured surveys by eliciting rich and contextualized responses, which are typically analyzed qualitatively. However, their potential contribution to quantitative measurement remains underexplored. In this paper, we introduce ConvScale, an AI-supported approach that transforms psychometric scales into natural conversational interviews while preserving the original measurement structure. Based on interview data, ConvScale predicts item-level scores and aggregates them to derive scale-based assessments. In a within-subjects study with 18 participants, our results show that ConvScale-derived scores align closely with participants' self-report scores at both the item and construct levels, while maintaining moderate internal reliability; however, the structural validity was inadequate. In light of this, we discussed the potential of supporting quantitative measurement through interviews and proposed implications for future designs.

We study Bogoliubov quasiparticle spectrum in a two-band system with momentum-dependent hybridization between a dispersive band and a flat band. The interplay between the interband mixing and intraband Cooper pairing may give rise to a parabolic node in the spectrum of flat band quasiparticles, resulting in a quadratic temperature dependence of the superconducting phase stiffness at low temperatures. We also comment that nonmagnetic disorder induces Machida-Shibata deep subgap resonances suggesting the sensitivity of flat-band superconductivity to disorder.

The density dependence of the nuclear symmetry energy remains a central open problem in nuclear physics. Parity violating electron scattering experiments have provided largely model-independent determinations of the neutron skin thickness of both 48Ca and 208Pb, whose consistent theoretical interpretation remains challenging. A new measurement of the matter radius of the unstable, doubly magic nucleus 132Sn provides an important additional constraint. Using a representative set of covariant energy density functionals spanning a wide range of isovector properties, we show that at least one of these models can simultaneously reproduce the charge and matter radii of132Sn. When interpreted together with the PREX and CREX results, the new measurement--much like CREX--favors a relatively soft symmetry energy. These findings underscore the need for an independent confirmation of the PREX result, such as the anticipated MREX campaign at the MESA facility.

We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian for arbitrary perturbation strengths, and explicit finite-time error bounds, that quantify leakage from the unperturbed center sector, are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.

We report the first measurement of the underground cosmic muon flux at the Stawell Underground Physics Laboratory. The measurement uses eight EJ200 plastic scintillator panels, equipped with Hamamatsu R13089 PMT pairs at the ends, which are the primary components of the muon veto system for the upcoming SABRE South experiment. The measured muon flux is f = (6.33 +/- 0.04_stat +/- 0.35_sys) x 10^{-8} [s^{-1} cm^{-2}]. This measurement is in excellent agreement with simulations, with a relative uncertainty an order of magnitude smaller than the modelling uncertainty.

For a positive integer $k$, let \[ σ_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$, and let $ν_p(m)$ denote the $p$-adic valuation of an integer $m$. Motivated by recent work on the $p$-adic valuation of $σ_k(n)$, we study $ν_2(σ_k(n))$ in detail. We prove that, for every integer $n\ge 2$, \[ ν_2(σ_k(n)) \le \begin{cases} \lceil \log_2 n \rceil, & \text{if $k$ is odd},\\[1mm] \lfloor \log_2 n \rfloor, & \text{if $k$ is even}. \end{cases} \] These bounds are best possible. More precisely, if $k$ is odd, then equality holds if and only if $n$ is a product of distinct Mersenne primes; if $k$ is even, then equality holds if and only if $n=3$. We also obtain an explicit formula for $ν_2(σ_k(n))$ in terms of the prime factorization of $n$.

Batteries play a critical role in microgrid energy management by ensuring power balance, enhancing renewable utilization, and reducing operational costs. However, battery degradation poses a significant challenge, particularly under extreme temperatures. This paper investigates the optimal trade-off between battery degradation and operational costs in microgrid dispatch to find a robust cost-effective strategy from a full life-cycle perspective. A key challenge arises from the endogenous uncertainty (or decision-dependent uncertainty, DDU) of battery degradation: Dispatch decisions influence the probability distribution of battery degradation, while in turn degradation changes battery operation model and thus affects dispatch. In this paper, we first develop an XGBoost-based probabilistic degradation model trained on experimental data across varying temperature conditions. We then formulate a parametric model predictive control (MPC) framework for microgrid dispatch, where the weight parameters of the battery degradation penalty terms are tuned through long-term simulation of degradation and dispatch interactions. Case studies validate the effectiveness of the proposed approach.

Hybrid work settings often lack the informal communication that naturally emerges from spontaneous encounters and ambient awareness of coworkers' activities, potentially hindering team collaboration. To address this challenge, we explored how lightweight interactions can be integrated into awareness-supporting technologies for fostering informal communication. Our experiential design approach focused on how information is perceived and processed rather than explicit content exchange. Through brainstorming, speculating, and prototyping, we explored the design space for small hybrid teams. By annotating and analyzing design concepts, speculative scenarios, and prototypes, we developed a framework that identified design options for lightweight interactions and methods for integrating them with information displays.

Achieving efficient three-dimensional feedback cooling of levitated nanoparticles is a key requirement for precision sensing and quantum control in levitated optomechanics. Here we demonstrate three-dimensional optical feedback cooling of a levitated nanoparticle using an interference-enhanced optical force generated within a single beam path. In this scheme, a weak auxiliary field co-propagates with the trapping tweezer and interferes with it to produce a tunable optical force that enables cold damping along all three center-of-mass motional axes without additional beam paths or trap reconfiguration. Using this approach, we cool a 142-nm-diameter silica nanoparticle in high vacuum to effective temperatures of 625.8, 711.6, and 19.9 mK along the $x$, $y$, and $z$ directions, respectively, at a pressure of $8.5\times10^{-6}$ mbar. The cooling dynamics and their dependence on feedback gain and pressure are well described by a cold-damping model. Because the feedback force is generated optically, the scheme does not rely on electrical actuation and is directly compatible with neutral particles. These results establish interference-based optical forces as a simple and broadly applicable mechanism for three-dimensional feedback control in levitated optomechanics, with a clear pathway toward the quantum regime under improved vacuum and detection conditions.

High-energy colliders, exemplified by the CERN's Large Hadron Collider (LHC), constitute genuine quantum machines. In alignment with Richard Feynman's foundational vision for quantum computing, collider physics emerge therefore as a prime candidate for quantum simulations. Prospective applications include Quantum Machine Learning for collider data analysis, accelerated evaluation of complex multiloop Feynman diagrams, efficient jet clustering, enhanced parton shower simulations, and related computational challenges. We discuss two specific applications: the identification of causal structures in multiloop vacuum amplitudes, a fundamental component of the Loop-Tree Duality exhibiting deep connections to graph theory; and high-dimensional function integration and sampling. The latter constitutes an initial step toward realizing a fully fleged quantum event generator capable of operating at high perturbative orders.

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of cohomological operations, partially linked to some homological operations. We treat here both the case of a specific class of precubical sets and of general directed spaces.

This paper develops an asymptotically efficient recursive identification algorithm for autoregressive systems with exogenous inputs under one-bit communications. In particular, the proposed method asymptotically achieves the Cramer-Rao lower bound (CRLB) based on the original data before quantization (original CRLB), whereas existing approaches typically attain only the CRLB corresponding to the quantized observations. The primary reason is that the existing methods quantize only the current system output, resulting in non-negligible information loss under one-bit quantization. To overcome this challenge, we present a novel quantization method that integrates both current and historical system outputs and inputs to provide richer parameter information in one-bit data, allowing the information loss caused by quantization to become a minor term relative to the original CRLB. Based on this technique, a corresponding remote estimation algorithm is further proposed. To address the convergence analysis challenge posed by the non-independence of the one-bit data, we establish a new framework that analyzes the tail probability of integrated data formed by combining current and historical system outputs and inputs before quantization, thereby eliminating the need for the traditional independence assumption on the quantized data. It is proven that the remote estimate achieves asymptotic normality, and the error covariance matrix converges to the original CRLB, confirming its asymptotic efficiency. Compared to existing identification algorithms under one-bit data, this method reduces the asymptotic mean squared error by at least 36%. Several numerical examples are simulated to show the effectiveness of the proposed algorithm.

This work presents a new design for broadband absorption of low-frequency acoustic waves using a thin coating made of subwavelength acoustic resonators arranged periodically on a reflective surface. We first study the associated scattering problem and the corresponding subwavelength resonance problem, and then derive analytical approximations for the resonant frequencies and the reflection coefficient in terms of the periodic capacitance matrix in a half-space with a Dirichlet boundary condition. These approximations yield an effective macroscopic description of the coating via an impedance boundary condition and clarify the mechanism of superabsorption through an approximate coupling condition. Moreover, they lead to a reduced order model that enables efficient evaluation of the scattered waves over a frequency band and accelerates broadband absorption design. Building on this reduced order model, we develop a gradient based shape optimization method using shape derivatives of the resonant quantities to achieve broadband absorption. Numerical experiments demonstrate the broadband performance and the effectiveness of the proposed design procedure.

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg 0.$$ We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of $M/I^{n} M$ as a function of $n$, when $R$ is $\mathbb{N}$-graded, $I$ is homogeneous, and $M$ is $\mathbb{Z}$-graded. When $I$ is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of $M/I^{n} M$ either coincides with that of the colon submodule $(0 :_{M} I)$, or is a polynomial in $n$ of degree one whose leading coefficient is one of the degrees of the generators of $I$. This dichotomy depends exclusively on two cases determined by $(0:_{M} I)$. Thus, we recover and considerably strengthen the main results of Fiorindo-Ghosh [Nagoya Math. J. 258 (2025), 296-310.], where asymptotic linearity was shown under the additional assumption that $(0:_{M} I)=0$.

In this work, a novel approach to Bayesian model calibration routines is developed which reinterprets the traditional definition of model discrepancy as defined by Kennedy and O'Hagan (KOH). The novelty lies in the integration of $δ_θ(x_i)$ GPs within the simulator, which is approximated as a GP surrogate model to ensure computational tractability. This approach assumes that the utilized simulator sufficiently predicts observed trends when calibrated with respect to the application domain, and that all model-form errors can be attributed to uncertainty in the input parameters. In contrast, the KOH method assumes discrepancy to be inherently decoupled from the simulator, acting as a 'catch-all' for various sources of model error. The new method is applied to Molecular Dynamics observations of the critical stress to drive dislocation dipoles, and equivalent predictions using a Discrete Dislocation Dynamics simulator whose coarse-grained physical interpretation of the underlying physical mechanisms requires calibration against MD observations. We present an overview of similar state-aware calibration routines; differentiate the provided approach through redefining the commonly used discrepancy Gaussian process and benchmark against KOH. A philosophical argument as to when application of the proposed method is appropriate is provided, and future directions for expanding upon this methodology are proposed.

Near-field propagation in extremely large-scale MIMO (XL-MIMO) enlarges the beam training (BT) search space by introducing an additional range dimension, which makes conventional codebook-based beam sweeping prohibitively expensive under limited pilot resources, especially for multiuser sub-connected hybrid architectures. This letter proposes a deep-learning-based interference-aware multiuser BT framework (DL-IABT) that directly predicts analog beam indices from a small number of uplink sensing measurements. By exploiting a subarray-level approximation, a far-field codebook is adopted to represent each subarray response with negligible mismatch. To enable end-to-end (E2E) learning, we derive a variant-MSE surrogate loss by eliminating the digital precoder through a closed-form MMSE solution from KKT conditions, which implicitly accounts for multiuser interference (MUI). The proposed network integrates a complex-valued sensing front-end, a shared complex-valued encoder, a Transformer-based multiuser predictor, and a scalable Gumbel--Softmax beam selection head. Simulation results show that DL-IABT achieves near-optimal sum-rate performance while providing markedly higher effective throughput under pilot overhead constraints.

We introduce a new game played on graphs, ``Agents and Adversary". This game is reminiscent of ``Cops and Robbers" but has some fundamental differences. We classify infinite families of graphs as Agents-win and Adversary-win. We then define a new type of graph symmetry and use it to define a winning strategy for Adversary. Finally, we give tight upper and lower bounds for Agents' time-to-win on several infinite families of graphs.

A flat quadratic quasi-Frobenius Lie superalgebra is a quadratic Lie superalgebra equipped with an additional symplectic structure that is flat with respect to the natural symplectic product. In this paper, we introduce the notion of a flat quadratic double extension of a flat quadratic quasi-Frobenius Lie superalgebra, in the cases where both the symplectic structure and the quadratic structure are either even or odd. We show that, over an algebraically closed field, any such Lie superalgebra can be constructed through a sequence of flat quadratic double extensions starting from the trivial algebra $\{0\}$. Moreover, when the quadratic and symplectic structures have different parity, we introduce the notion of a planar double extension, which constitutes the main novelty of this paper. In this case, we prove that such Lie superalgebras have total dimension $4n$. Finally, we classify flat quadratic quasi-Frobenius Lie superalgebras of dimension at most four and present explicit examples in dimensions six and eight.

A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S|that contains a representation of each element in S exactly once as a substring. If S is the set of k-subsets of [n] = {1, 2, . . . , n}, it is well-known that universal cycles do not always exists when applying a simple string representation, where 12 or 21 could represent the subset {1, 2}. Similarly, if S is the set of k-multisets of [n], it is also known that universal cycles do not always exist using a similar representation, where 112, 121, or 211 could represent the multiset {1, 1, 2}. By mapping these sets to an appropriate family of labeled graphs, universal cycles are known to exist, but without a known efficient construction. In this paper we consider a new representation for k-subsets and k-multisets that leads to efficient universal cycle constructions for all n, k >=2. We provide successor-rule algorithms to construct such universal cycles in O(n) time per symbol using O(n) space and demonstrate that necklace concatenation algorithms allow the same sequences to be generated in O(1) amortized time per symbol. They are the first known efficient universal cycle constructions for k-multisets. The results are obtained by considering constructions for bounded-weight de Bruijn sequences. In particular, we demonstrate that a bounded-weight generalization of the Grandmama de Bruijn sequence can be constructed in O(1) amortized time per symbol.

In this work, we present a mixed precision algorithm that leverages the Gram matrix and Jacobi methods to compute the singular value decomposition (SVD) of tall-and-skinny matrices. By constructing the Gram matrix in higher precision and coupling it with a Jacobi algorithm, our theoretical analysis and numerical experiments both indicate that the singular values computed by this mixed precision thin SVD algorithm attain high relative accuracy. In practice, our mixed precision thin SVD algorithm yields speedups of over 10x on a single CPU and about 2x on distributed memory systems when compared with traditional thin SVD methods.

We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a $3\times 3$ Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.

Sixth-generation (6G) wireless networks are expected to support autonomous, immersive, and mission-critical services that require not only extreme data rates and ultra-low latency but also adaptive reasoning, cross-domain coordination, and objective-driven control across distributed edge-cloud infrastructures. Current AI-enabled network management remains largely data-centric, relying on discriminative models that optimize intermediate quality-of-service metrics without explicitly reasoning about long-term service objectives. This article advocates a transition from bit-centric communication toward knowledge-centric coordination in 6G systems. Semantic communication prioritizes task-relevant information and contextual meaning over raw data delivery, while generative artificial intelligence enables predictive reasoning and adaptive policy synthesis aligned with dynamic service intents. Network optimization is therefore reframed around goal-oriented performance metrics capturing application-level outcomes rather than solely protocol-level indicators. To operationalize this vision, we introduce Kraken, a multi-agent architecture composed of a Knowledge Plane, a distributed Agent Plane, and a semantic-aware Infrastructure Plane. By integrating semantic communication, generative reasoning, and goal-oriented optimization over a shared knowledge substrate, Kraken enables scalable collective intelligence and outlines an evolutionary path from current 5G infrastructures toward knowledge-native 6G systems.

High-voltage direct current (HVDC) technology has played a crucial role for long-distance transmission of renewable power generation. However, the integration of large-capacity HVDC lines introduces significant frequency security challenges during HVDC fault emergencies. This paper proposes an emergency-aware and frequency-constrained HVDC planning method to optimize the capacity of inter-area HVDC tie-lines in a multi-area asynchronously interconnected grid. Firstly, a coordinated emergency frequency control scheme is proposed to allocate the emergency control resources during HVDC faults. Then, an enhanced system frequency response model integrating event-driven emergency frequency control is developed and a weighted oblique decision tree approach is employed to extract frequency nadir security constraints. The proposed planning model considers all potential HVDC fault emergencies while treating candidate HVDC capacities as decision variables. Simulation results demonstrate superior performance in balancing economic efficiency with frequency security requirements, providing a practical solution for inter-area HVDC planning.

Quantum teleportation uses a shared entangled resource, local operations, and a digitally error-corrected classical channel to transfer quantum states between distant parties. We introduce a hybrid teleportation-direct transmission protocol for state transfer that still exploits entanglement, but replaces classical communication and digital error correction with an analog feedforward through a noisy quantum channel. We show that quantum teleportation outperforms this protocol if the communication channel reduces the entanglement of all bipartite states having the same amount of entanglement as the resource; otherwise, the hybrid protocol is optimal. We apply our result to the state transfer of a uniformly distributed coherent-states codebook, highlighting experimentally relevant scenarios where our protocol is most effective. Our findings are directly relevant to both optical and superconducting microwave channels, where analog feedforward techniques have been recently implemented.

We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be palindromic and to have non-negative integer coefficients with a combinatorial interpretation. Along the way, we uncover unexpected symmetries between the values of the zeta function at negative and positive integers, expressed at the level of their generating functions. Using these symmetries, we ultimately establish a functional equation of the type \( s \longleftrightarrow 1-s \) for a natural completion of the zeta function.

Quantum computers have demonstrated utility in simulating quantum systems beyond brute-force classical approaches. As the community builds on these demonstrations to explore using quantum computing for applied research, algorithms and workflows have emerged that require leveraging both quantum computers and classical high-performance computing (HPC) systems to scale applications, especially in chemistry and materials, beyond what either system can simulate alone. Today, these disparate systems operate in isolation, forcing users to manually orchestrate workloads, coordinate job scheduling, and transfer data between systems -- a cumbersome process that hinders productivity and severely limits rapid algorithmic exploration. These challenges motivate the need for flexible and high-performance Quantum-Centric Supercomputing (QCSC) systems that integrate Quantum Processing Units (QPUs), Graphics Processing Units (GPUs), and Central Processing Units (CPUs) to accelerate discovery of such algorithms across applications. These systems will be co-designed across quantum and classical HPC infrastructure, middleware, and application layers to accelerate the adoption of quantum computing for solving critical computational problems. We envision QCSC evolution through three distinct phases: (1) quantum systems as specialized compute offload engines within existing HPC complexes; (2) heterogeneous quantum and classical HPC systems coupled through advanced middleware, enabling seamless execution of hybrid quantum-classical algorithms; and (3) fully co-designed heterogeneous quantum-HPC systems for hybrid computational workflows. This article presents a reference architecture and roadmap for these QCSC systems.

The expansion of Internet of Things (IoT) devices has increased the attack surface of networks, necessitating a robust and adaptive intrusion detection systems. Machine learning based systems have been considered promising in enhancing the detection performance. Federated learning settings enabled us to train models from network intrusion data collected from clients in a privacy preserving manner. However, the effectiveness of these systems can degrade over time due to concept drift, where patterns in data evolve as attackers develop new techniques. Realistic detection models should be non-stationary, so they can be continuously updated with new intrusion data while maintaining their detection capability for older data. As IoT environments are resource constrained, updates should consume minimal computational resources. This study provides a comprehensive performance analysis of incremental federated learning in enhancing the long term performance of non stationary IDS models in IoT networks. Specifically, we propose LSTM models within a federated learning setting to evaluate incremental learning approaches that utilize data and model-based measures against catastrophic learning under drift conditions. Using the CICIoMT2024 dataset, which includes various attack variants across five major categories, we conduct both binary and multiclass classification to provide a granular analysis of the intrusion detection task. Our results show that cumulative incremental learning and representative learning provide the most stable performance under drift, while retention-based methods offer a strong accuracy and latency trade off. The study offers new insights into the interplay between training strategy performance and latency in dynamic IoT environments, aiming to inform the development of more resilient IDS solutions considering the resource constraints in IoT devices.

Competing processes shape Europa's surface: geological activity replenishes material through resurfacing, while bombardment by charged particles alters surface chemical composition. Each process leaves distinct spectral signatures. We present a novel data-driven analysis of JWST NIRSpec-IFU observations of Europa's leading hemisphere across three observing geometries, targeting nine spectral bands sensitive to water ice, radiolytic products, and volatiles. Through spectral factorization, we isolate the dominant components of spectral variability and reconstruct their spatial distributions. We find that CO2 enrichment extends beyond Tara Regio, and covers multiple chaos units in a lens-like pattern. These CO2-enriched areas co-occur with anomalous ice-texture signatures. Together, these findings suggest that enrichment in volatiles on Europa may reflect retention-favorable near-surface microphysics as well as emplacement, refining how they are interpreted in the context of surface--interior exchange. This has implications for interpreting the sources and supply rates of extant carbon-bearing species and, ultimately, for assessing Europa's habitability.

We consider the singular elliptic problem of the form \[ -Δu + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural spectral assumptions on $-Δ+V$, geometric assumptions on the manifold $M$ ensuring the Sobolev embedding $H^1(M)\hookrightarrow L^{2^*}(M)$, and a suitable global integrability/smallness condition involving $\mathcal{A}$, $\mathcal{B}$, and a function $ψ\in H^1(M)$, we prove the existence of a nonnegative finite-energy supersolution. If, in addition, the Ricci curvature is nonnegative and $\mathcal{B}\ge 0$, we obtain a positive finite-energy solution. The proof relies on a family of $\varepsilon$-regularized problems, mountain pass arguments, and a limiting procedure in which Harnack's inequality plays a crucial role in handling the singular term on noncompact manifolds. We also prove a nonexistence result showing that the global integrability condition on $\mathcal{A}$ is, in a precise sense, necessary for the existence of nonnegative supersolutions.